Instant Notes: Analytical Chemistry

Example

A calibration graph was prepared as part of a validation procedure for a new

method to determine an active constituent of a sun cream by UV spectrophoto-

metry. The following data were obtained:

Analyte conc. (mg cm-^3 ) 0 20 40 60 80 100 120

UV absorbance at 325 nm 0.095 0.227 0.409 0.573 0.786 0.955 1.123

The data is first checked for linearity by calculation of the correlation coefficient,

r, and visual inspection of a plotted curve. Some calculators and computer soft-

ware can perform the computation from the raw data, but it is instructive to

show the full working, for which tabulation is preferable.

xi yi (xi−x

_

)(xi−x

_

)^2 (yi−y

_

)(yi−y

_

)^2 (xi−x

_

)(yi−y

_

)

0 0.095 − 60 3600 −0.5004 0.2504 30.024

20 0.227 − 40 1600 −0.3684 0.1357 14.736

40 0.409 − 20 400 −0.1864 0.0347 3.728

60 0.573 0 0 −0.0224 0.0005 0

80 0.786 20 400 0.1906 0.0363 3.812

100 0.955 40 1600 0.3596 0.1293 14.384

120 1.123 60 3600 0.5276 0.2784 31.656

S 420 4.168 0 11200 0 0.8653 98.340

x

_

= 60 y

_

=0.59543

Substitution of the totals in columns 4, 6 and 7 in equation (2) gives

r=98.340/(11200 ¥0.8653)1/2=98.340/98.445 =0.9989

Figure 3and the correlation coefficient of 0.9989 show that there is a good linear

relation between the measured UV absorbance and the analyte concentration.

The slope and y-axis intercept of the regression line, given by equations (3)

and (4) respectively are

b=98.340/11200 =0.00878 a=0.59543 -(0.00878 ¥60) =0.0686

The y-axis intercept, slope and analyte masses or concentrations calculated

by interpolation from the regression line are all affected by errors. Additional

equations can be used to obtain the following statistics:

● estimated standard deviations for the slope and intercept;

● estimated standard deviations for analyte masses or concentrations deter-

mined from the calibration graph;

● confidence limits for analyte masses and concentrations at selected proba-

bility levels;

● limit of detection of the analyte (vide infra).

Confidence limits (Topic B2) over the entire range of the calibration graph at

selected probability levels, e.g. 95 or 99 percent, can be displayed (dashed

curves, Fig. 3). A horizontal line drawn through a given experimental point on

the regression line and intersecting the confidence limits lines on either side

gives the upper and lower limits for that particular mass or concentration.

Figure 3 shows the 99% limits, the narrowest interval being at the centroid, x

_

, y

_

,

of the graph, and widening steadily towards each end.

Some calculators and computer packages have the ability to perform the

regression calculations described. Where there is a nonlinear relation between

B4 – Calibration and linear regression 45